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contrapositions

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Possible Dictionary Clues
Plural form of contraposition.
Conversion of a proposition from all A is B to all not-B is not-A.

Contrapositions might refer to
In logic, Contraposition is an inference that says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped: the contrapositive of * * * * P * → * Q * * * {\displaystyle P\rightarrow Q} * is thus * * * * ¬ * Q * → * ¬ * P * * * {\displaystyle \neg Q\rightarrow \neg P} * . For instance, the proposition "All cats are mammals" can be restated as the conditional "If something is a cat, then it is a mammal". Now, the law says that statement is identical to the contrapositive "If something is not a mammal, then it is not a cat." * The contrapositive can be compared with three other relationships between conditional statements:* Inversion (the inverse), * * * * ¬ * P * → * ¬ * Q * * * {\displaystyle \neg P\rightarrow \neg Q} * * "If something is not a cat, then it is not a mammal." Unlike the contrapositive, the inverse's truth value is not at all dependent on whether or not the original proposition was true, as evidenced here. The inverse here is clearly not true. * Conversion (the converse), * * * * Q * → * P * * * {\displaystyle Q\rightarrow P} * * "If something is a mammal, then it is a cat." The converse is actually the contrapositive of the inverse and so always has the same truth value as the inverse, which is not necessarily the same as that of the original proposition. * Negation, * * * * ¬ * ( * P * → * Q * ) * * * {\displaystyle \neg (P\rightarrow Q)} * * "There exists a cat that is not a mammal. " If the negation is true, the original proposition (and by extension the contrapositive) is false. Here, of course, the negation is false.Note that if * * * * P * → * Q * * * {\displaystyle P\rightarrow Q} * is true and we are given that Q is false, * * * * ¬ * Q * * * {\displaystyle \neg Q} * , it can logically be concluded that P must be false, * * * * ¬ * P * * * {\displaystyle \neg P} * . This is often called the law of contrapositive, or the modus tollens rule of inference.

Contrapositions might refer to

In logic, Contraposition is an inference that says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped: the contrapositive of
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* is thus
*
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* ¬
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* {\displaystyle \neg Q\rightarrow \neg P}
* . For instance, the proposition "All cats are mammals" can be restated as the conditional "If something is a cat, then it is a mammal". Now, the law says that statement is identical to the contrapositive "If something is not a mammal, then it is not a cat."
* The contrapositive can be compared with three other relationships between conditional statements:* Inversion (the inverse),
*
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*
* {\displaystyle \neg P\rightarrow \neg Q}
*
* "If something is not a cat, then it is not a mammal." Unlike the contrapositive, the inverse's truth value is not at all dependent on whether or not the original proposition was true, as evidenced here. The inverse here is clearly not true.
* Conversion (the converse),
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* {\displaystyle Q\rightarrow P}
*
* "If something is a mammal, then it is a cat." The converse is actually the contrapositive of the inverse and so always has the same truth value as the inverse, which is not necessarily the same as that of the original proposition.
* Negation,
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* {\displaystyle \neg (P\rightarrow Q)}
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* "There exists a cat that is not a mammal. " If the negation is true, the original proposition (and by extension the contrapositive) is false. Here, of course, the negation is false.Note that if
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* is true and we are given that Q is false,
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* , it can logically be concluded that P must be false,
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* {\displaystyle \neg P}
* . This is often called the law of contrapositive, or the modus tollens rule of inference.

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